On the square root of the inverse different

Abstract Let $F_{\pi }$ be a finite Galois-algebra extension of number field F , with group G . Suppose that }/F$ is weakly ramified and the square root $A_\pi $ inverse different $\mathfrak {D}_{\pi }^{-1}$ defined. (This latter condition holds if, for example, $|G|$ odd.) Erez has conjectured class $(A_\pi )$ in locally free $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ $\mathbf G$ equal to Cassou–Noguès–Fröhlich $W(F_{\pi }/F)$ associated $F_\pi /F$ This conjecture been verified many cases. We establish precise formula terms all cases where defined tame, are thereby able deduce that, general, not $W(F_\pi /F)$